Blur is an important attribute of human spatial vision, and sensitivity to blur has been the subject of considerable experimental research and theoretical modeling. Often, these models have invoked specialized concepts or mechanisms, such as intrinsic blur, multiple spatial frequency channels, or blur estimation units. In this paper, we review the several experimental studies of blur discrimination and find that they are in broad empirical agreement. However, contrary to previous modeling efforts, we find that specialized mechanisms are not required and that the essential features of blur discrimination are fully accounted for by a visible contrast energy (ViCE) model, in which two spatial patterns are distinguished when the integrated difference between their masked local visible contrast energy responses reaches a threshold value. In the ViCE model, intrinsic blur is represented by the high-frequency limb of the contrast sensitivity function, but the low-frequency limb also contributes to the predictions for large reference blurs, and the model includes masking, which improves predictions for high-contrast stimuli.

*blur*is to render indistinct, usually by smudging or smearing, but sometimes by dimming, or through obscuration by fog, by soot, or by a blow to the head. In the narrower technical context of vision, optics, and imaging, blurring generally connotes a smearing of an image, through some amount of low-pass filtering.

*x*-axis. Where possible, a consistent color code is used to identify each study. From each study, we have derived a summary curve by averaging over observers or conditions, as described in the text. These summaries will ease comparison of data and models.

^{1}We plot their results below in Figure 4. These data again show a prominent dipper shape, with a detection threshold of about 0.6 arcmin and a minimum at a reference blur just above 1 arcmin.

*e*Gaussian half-widths). They also found that contrasts above 0.3 produced little or no change in thresholds, but that contrasts of 0.05 and 0.02 did produce appreciably higher thresholds.

^{2}. In general, the results resemble those of Mather and Smith (2002) and Wuerger et al. (2001). As a summary record, we use the average of their three highest luminances.

Study | Method | Duration (ms) | Blur | Edge extent (arcmin) | Contrast | Mean (cd/m^{2}) | Observers |
---|---|---|---|---|---|---|---|

Chen | 2IFC | 200 | Gaussian | 40 | 1 | 0.26, 2.58, 5.16, 25.8 | 4 |

Hamerly | 2AFC | ∞? | Gaussian | 54 | 0.33, 0.82 | 86 | 2 |

Hess | 3AFC, 2IFC | ∞ | Cosine | 39 | 0.3 | 500 | 2 |

Mather | 2IFC | 500 | Gaussian | 262 | 0.1, 0.2, 0.4, 0.6, 0.68, 0.8 | 37.5 | 5 |

Pääkkönen | 2IFC | 150 | Gaussian | 20 | 0.35 | 43.7 | 3 |

Watt | 2AFC | ∞ | Gaussian, ramp, cosine | 12 | 0.8 | 292 | 2 |

Wuerger | 2IFC | 1000 | Gaussian | 240 | 0.1 | 40 | 4 |

Westheimer | 2AFC | 250 | Ramp | 30 | Various | 5 |

*ρ*= 1). A second increment, dependent upon contrast, that reflects the accuracy with which individual blurs can be estimated is added to this simple Weber model. The final prediction is that threshold is the quadratic sum of the two increments. A good fit to data is shown, but parameters are not given. For reference, and converting to the notation of this paper, the resulting formula for the TVR is

*k*

_{L}is a parameter.

*ρ*= 1, find a good fit, and estimate parameters for their conditions. For luminance blur at 0.1 contrast, they found internal blur

*β*of about 1.2 arcmin and a Weber constant

*ω*of about 1.2. They also consider a model based on contrast sensitivity, which is, in many respects, the same as the visible contrast energy model that we discuss below. They compute amplitude spectra of the two blurred edges to be distinguished, weight by the contrast sensitivity function, and inverse Fourier transform; take their difference; compute the RMS contrast of the result; and divide that by the RMS contrast of the filtered reference image. This formula differs in several respects from ours, but the most significant difference in this context is that their model lacks a parameter to control the contrast normalization. Some details of their model are not clear from the description. In any case, for this model, they state that: “For higher reference blurs, a single channel model does not predict blur difference thresholds.” It is unclear why their formulation fails to account for the blur discrimination function, while ours succeeds, as shown below.

*ρ*is 1. Here, we allow it to vary. As we shall see, values slightly greater than one fit the data better than one and yield a slight upward concavity of the TVR curve. Likewise, in a traditional Weber model, the so-called Weber fraction is

*ω*− 1. Although any consistent measure of blur could be used, here we assume that blur is measured by the standard deviation of the equivalent Gaussian blur.

*β*and the external image blur. Because the blurs are conceived as the result of successive convolutions, the blurs combine as the square root of the sum of their squares. In the case of the smaller reference blur,

*r,*this result is

*β*shifts the early part of the curve vertically, without much effect on the portion beyond the dip. The Weber ratio

*ω*shifts the entire curve vertically. Increasing

*ρ*lowers the early part of the curve and lifts the later part of the curve. It also increases the upward curvature of the later part of the curve.

_{10}domain. Parameter estimates are shown in Table 2, along with RMS errors. We show estimates and error for both

*ρ*free and

*ρ*= 1. In Figure 12, we show plots of individual data sets and fitted curves for the case of

*ρ*free. Considering the RMS values and the plots in Figure 12, it is evident that the model fits quite well to the seven data sets.

Study | ρ free | ρ = 1 | |||||
---|---|---|---|---|---|---|---|

β | ω | ρ | RMS | β | ω | RMS | |

Chen | 1.20 | 1.18 | 1.05 | 0.043 | 0.937 | 1.29 | 0.055 |

Hamerly | 0.39 | 1.07 | 1.00 | 0.071 | 0.389 | 1.07 | 0.071 |

Hess | 1.71 | 1.06 | 1.03 | 0.034 | 1.110 | 1.14 | 0.074 |

Mather | 1.54 | 1.13 | 1.04 | 0.040 | 1.230 | 1.21 | 0.046 |

Pääkkönen | 0.78 | 1.13 | 1.01 | 0.036 | 0.744 | 1.14 | 0.036 |

Watt | 1.56 | 1.00 | 1.06 | 0.055 | 0.607 | 1.11 | 0.098 |

Wuerger | 1.91 | 1.00 | 1.13 | 0.045 | 1.170 | 1.21 | 0.052 |

*ρ*= 1 and

*ρ*free, we see that allowing

*ρ*to vary lowers the error substantially only in those cases where large reference blurs are used. This is expected, since the effect of

*ρ*> 1 is to bend the curve upward at higher reference blurs.

*ρ*free, the intrinsic blur parameters vary between 0.39 (Hamerly) and 1.91 arcmin (Wuerger) and reflect the large variation in absolute sensitivity in the data. The Weber ratios vary between 1 and 1.18. Values of

*ρ*vary between 1 and 1.13. Because the three parameters are somewhat correlated in their effects, it is difficult to interpret absolute values for the various parameters. When

*ρ*= 1, values of

*β*are generally, sometimes substantially, lower, while values of

*ω*may be substantially different, sometimes higher, sometimes lower. When

*ρ*= 1, the parameters are less correlated, and absolute values may have more meaning. With the obvious exception of Hamerly, the values of

*β*are near to 1, and the values of

*ω*are around 1.18.

*r*+

*a*) and then combines this with a constant intrinsic blur (

*β*) to obtain perceived blur (

*σ*). The lack of an explicit algorithm to compute perceived blur from the image renders this model incapable of dealing with edges with non-Gaussian blur or with changes in contrast.

*s*is given by

*s*and an integral of 1. The scale

*s*is an alternative to the standard deviation as a measure of the Gaussian width. The two are related by

*s*. It is the Fourier transform of

*G*(

*x, s*). An attractive feature of this parameterization of Gaussians is that the Fourier transform of a Gaussian of scale

*s*(in degrees) is a Gaussian of scale 1/

*s*(in cycles/degree).

*r*′ and

*r*′ +

*a*′, respectively:

*ϕ*and surround scale

*θ,*with a surround weight of

*λ*and gain

*γ*. The parameter

*λ*lies between 0 and 1 and represents the ratio of areas of the two Gaussians:

*c,*the visible contrast energy difference can thus be written as

*λ,*introduced above, determines the attenuation at low spatial frequencies.

*a*′,

*r*′,

*ϕ,*and

*θ*are Gaussian scales expressed in degrees, and

*V*is the visible contrast energy:

*V*= 1. In order to generate predictions for this model, we require values for the parameters

*ϕ, θ, γ,*and

*λ*. We have obtained these values through an approximate fit to a set of eleven contrast thresholds for Gabor functions from the ModelFest experiment (Carney et al., 2000; Watson & Ahumada, 2005). All of these targets employed a Gaussian aperture with a standard deviation of 0.5 deg. The first target, with a nominal spatial frequency of 0, was actually a simple Gaussian, with no sinusoidal modulation. That target is useful for estimating the sensitivity to very low spatial frequencies.

*γ*= 160.05,

*λ*= 0.7329,

*ϕ*= 2.456 arcmin, and

*θ*= 24.75 arcmin.

^{2}

*r,*we can solve for the value of added blur

*a*that yields

*V*= 1. A result is shown in Figure 15 for an edge contrast of

*c*= 0.2. This figure shows that the simplest visible contrast energy model, with no adjustment of parameters, predicts the essential features of the blur discrimination TVR function: the absolute threshold, the dipper, the location of the dip, the magnitude of the dip, and the rise at higher reference blurs.

*γ*generally shifts the predictions vertically,

*ϕ*vertically shifts the portion of the curve to the left of the dipper,

*θ*vertically shifts the portion to the right of the dipper, while

*λ*controls the slope of the curve to the right of the dipper. In the following section, we will provide a general explanation for these behaviors.

*a*(0.5 arcmin) and several values of reference blur

*r*. This difference spectrum, multiplied by the filter (orange curve) and squared, is the dashed curve, and it is the integral of this (the gray area) that must equal 1 at threshold. When

*r*is zero, the difference spectrum is restricted to high spatial frequencies, and little of it passes through the filter. When

*r*= 1 arcmin, the difference spectrum is concentrated at middle frequencies, and much more passes through the filter. Hence, a smaller

*a*is required to reach threshold. As

*r*increases to 8, the difference spectrum moves to still lower frequencies, where sensitivity is higher. However, because the width of the blur spectra (red and green) are inversely related to their kernel widths, their width difference diminishes, as does the width and magnitude of the difference spectrum. Hence, a larger

*a*is required to reach threshold. Put in other words, when the blur magnitudes are very small, the spectral region over which the spectra differ corresponds to a region with poor contrast sensitivity. Conversely, when blur magnitudes are very high, the bandwidth of the blurred stimuli is very small, as is that of their difference signal and hence its energy. Thus, at both ends of this range of blur, thresholds will be high but will be lower in the middle of this range. We also provide, as a demonstration, an animated version of this figure to allow the reader to explore other values of

*r, a,*and model parameters.

*l*

_{1}and

*l*

_{2}. Although it may be generalized to two spatial dimensions, we describe it here in only one dimension, which is sufficient to treat discrimination of one-dimensional edges.

*L*for the target luminance waveform and

*L*

_{0}for the preceding luminance waveform, then the space–time average luminance can be approximated by

*L*

_{a}

*κ*is a measure of the degree to which complete adaptation to the target has transpired. Although

*κ*should be a function of time, we imagine it to be slowly varying, at the time that observers must respond or by the time the target is extinguished, so we regard it as a constant here. The mixture of the two luminance waveforms is convolved with a kernel

*H*

_{s}(the “surround” kernel) that represents the spatial window over which the adapting average is estimated. The kernel has unit area, so that it computes a weighted average.

*H*

_{c}, also with unit area, that reflects optical and perhaps neural blurring. From this, the local average is subtracted, which also divides the result:

*L*

_{0}, in which case

*ϕ*(in degrees):

*τ*in units of contrast (0 <

*τ*< 1), squaring to compute energy, and then convolving with another Gaussian kernel

*H*

_{m}, with scale Ω (in degrees), that determines the neighborhood over which the energy is integrated. Here, we set Ω = 10/60 deg. Finally, we add one to the result and take the square root to yield the masking waveform, which then divides the contrast waveform to yield the masked contrast

*M*:

*τ*determines the approximate contrast at which masking begins to take effect.

*M*

_{1}and

*M*

_{2}that result from luminance waveforms

*L*

_{1}and

*L*

_{2}:

*γ*is a gain parameter,

*H*

_{p}is a Gaussian pooling kernel with scale

*δ*(in degrees), and

*p*

_{x}is the width of one waveform sample in degrees (this assumes a discrete sampled representation of the waveforms). The pooling kernel determines the area over which the energy is pooled. Based loosely on estimates from similar models fit to the ModelFest data, we set

*δ*= 1 deg. The Max operator returns the maximum of a waveform. Several of the steps involved in this model are illustrated in Figure 18.

*l*

_{1}and a constant waveform of the background luminance to create

*l*

_{2}. The result of this fit is shown in Figure 19. The estimated parameters are

*ϕ*= 2.766 arcmin,

*θ*= 21.6 arcmin,

*γ*= 217.65, and

*κ*= 0.772. Note that in this case (unlike Figure 14) we make use of the highest frequency Gabor and that the fit is considerably better, showing that the hyperbolic secant function is a much better representation of the center mechanism than the Gaussian.

*ϕ*= 2.758 arcmin,

*θ*= 18.76 arcmin,

*λ*= 0.78, and

*γ*= 329.9. The values are similar to those obtained here, but the gain is larger. However, the model in Watson and Ahumada was two-dimensional and included a Gaussian spatial aperture, an oblique effect, and two-dimensional CSF, all of which would entail a larger gain.

*λ*or

*κ*. Elsewhere, we have suggested that experimental artifacts in ModelFest may have exaggerated sensitivity to large Gaussian targets, artificially elevating sensitivity at low frequencies (Ahumada & Scharff, 2007). Since we used these data to estimate model parameters, this may explain the deviation at large reference blurs.

*τ*= 0.3. We did not attempt a precise estimate of this parameter, but values of 0.2 and 0.4 produced noticeably worse fits. The model may deviate from the data at the very lowest contrasts, possibly because at these very low contrasts, for human observers, the edge location becomes uncertain.

*ϕ*and

*θ*). It is well established that these increase with eccentricity (Graham, Robson, & Nachmias, 1978; Koenderink, Bouman, Mesquita, & Slappendel, 1978; Virsu & Rovamo, 1979; Watson, 1987). Hess et al. measured blur discrimination at several eccentricities and found shifts in the TVR that are at least roughly consistent with this idea. However, this variation in the CSF with eccentricity creates problems for some of the methods used in the studies reviewed here. For example, if a 2AFC method is used with a brief exposure (Westheimer et al., 1999), the observer cannot fixate both targets, and thus, the effective eccentricity of the edge is uncertain. Where the exposure duration is long, and fixation is uncontrolled (Hamerly & Dvorak, 1981; Hess et al., 1989; Watt & Morgan, 1983), we may imagine that observers move their eyes back and forth between test and reference edges, in which case the method is much like 2IFC, albeit with a brief interval. For accurate and consistent local measurements of blur discrimination, a 2IFC method with brief exposure and controlled fixation would appear to be required.

*ϕ,*which also determines sensitivity at high spatial frequencies and acuity. Thresholds, in general, depend upon the gain parameter

*γ,*which is analogous to peak contrast sensitivity. In this regard, it is worth noting that both contrast sensitivity and acuity decline with age in adults. From 1920s to 1980s, in the absence of obvious ocular pathology, contrast sensitivity at 16 cycles/deg declines by about 0.1 log unit per decade, while LogMAR acuity declines by 0.07 (Owsley, Sekuler, & Siemsen, 1983). Thus, we may expect substantial differences in blur sensitivity between observers as a result of differences in acuity and contrast sensitivity.

*ρ*= 1, excluding Hamerly & Dvorak, in Table 2). While this quantitative agreement may be reassuring, the mathematical reasons for it remain obscure, hidden in Equation 16. It should also be noted that the reasons for the eventual rise in thresholds at large reference blurs are very different for the two models. In the Weber model, it is because the model assumes that the discriminable difference in perceived blur rises in proportion to perceived blur, while in the ViCE model it is due largely to the attenuation of low spatial frequencies due to the surround mechanism.

*H*

_{c}and its scale

*ϕ*play a central role in the ViCE model in determining the sensitivity to blur. This kernel must incorporate any optical blur present in the eye of the observer. Any additional blur forms what is known as the neural transfer function (NTF). Sekiguchi, Williams, and Brainard (1993) estimated a very small additional blur, corresponding to an NTF represented by a Gaussian with a scale of 0.75 arcmin. Recently, we have argued that even this additional blur may be better attributed to limits on spatial summation, themselves the result of retinal inhomogeneity (Ahumada, Coletta, & Watson, 2010). Thus, most and perhaps all of the limits on blur discrimination, at least for small reference blurs, are due to optical limitations. However, discrimination at large reference blurs depends upon the behavior of the CSF at low spatial frequencies (see Figure 16), which is not optical in origin.

*f*(in cycles/degree) and Gaussian scale

*s*(in degrees) is given by

*ϕ*= 2.20027 arcmin,

*θ*= 25.2396 arcmin,

*λ*= 0.76, and

*γ*= 271. The values are similar to those obtained here, but the gain is larger. However, the model in Watson and Ahumada was two-dimensional and included a Gaussian spatial aperture, an oblique effect, and two-dimensional CSF, all of which would entail a larger gain.